2/6/2015 Puzzle:PattyPaperTrisection|Let'sPlayMath!
http://letsplaymath.net/2007/06/05/puzzlepattypapertrisection/ 1/4
Let'sPlayMath!
JUN52007
Puzzle:PattyPaperTrisection
[FeaturephotoabovebyMichaelCory(https://www.flickr.com/photos/khouri/6260501380/)viaFlickr(CCBY2.0).]
(http://letsplaymath.net/2007/06/05/puzzlepattypapertrisection/)
Oneofthegreatunsolvedproblemsofantiquity
(http://mathforum.org/dr.math/faq/faq.impossible.construct.html)wastotrisectanyangle(http://wwwgroups.dcs.standrews.ac.uk/~history/HistTopics/Trisecting_an_angle.html)usingonlythebasictoolsofEuclideangeometry:anunmarkedstraightedgeandacompass.Likethealchemistsdream(http://en.wikipedia.org/wiki/Alchemy)ofturningleadintogold,thisprovedtobeanimpossibletask.Ifyouwanttotrisectanangle,youhavetocheat.Astraightedgeandcompasscantdoit.Youhavetousesomesortofcrutch,justasanalchemistwouldhavetouseaparticleaccelerator(http://chemistry.about.com/cs/generalchemistry/a/aa050601a.htm)orsomething.
Onecheatthatworksistofoldyourpaper.Iwillshowyouhowitworks,andyourjobistoshowwhy.
Abouttheseads
Mathisagame,playingwithideas.
2/6/2015 Puzzle:PattyPaperTrisection|Let'sPlayMath!
http://letsplaymath.net/2007/06/05/puzzlepattypapertrisection/ 2/4
GatherYourMaterials
Apieceofpaperwithsquarecorners.(Iusedpattypaper.)Arulerorotherstraightedge.Apenorpencil.
OneStepataTime
Drawanyrandomangle(lessthan90degrees)withitsbaselinealongthebottomofyourpaperanditsvertexatthecorner.CallthevertexpointP.
(http://letsplaymath.files.wordpress.com/2007/06/trisection11.jpg)
Foldthepaperhorizontally,sothatpointPtouchesthesideofthepaperatsomeotherpointQ.DrawalinethroughQ,usingthefoldedupbaselineasyourguide.Unfoldthepaper,anddrawanotherlinealongthefold.Theselinesareshowninredonmypattypaper.
(Clicktheimageforalargerview.)
Canyouprovethatthesetwolinesareparalleltothebaselineofyourpaper?Canyoushowthattheymustbeequallyspaced?
(http://letsplaymath.files.wordpress.com/2007/06/trisection21.jpg)
2/6/2015 Puzzle:PattyPaperTrisection|Let'sPlayMath!
http://letsplaymath.net/2007/06/05/puzzlepattypapertrisection/ 3/4
Foldthepaper(dottedblueline)sothatpointPtouchesthelowerofthetwoparallellinesandpointQtouchesthetoplineofyouroriginalangle.ThesetouchpointsarelabeledPandQonthediagramabove.MarkpointPonyourpaper.
Unfoldthepaper.UseyourstraightedgetoconnectcornerPwithpointP.AlsodrawalinefromcornerPthroughthepointwherethisfoldmeetsthefirstfold.Thesetwolinesaremarkedingreenintheimage.
Yourangleisnowtrisected.Thelasttwolinesyoudrewcuttheoriginalangleinto3smallerangles,allequal.
CanYouProveIt?
Foldingpaperistheeasypart,butnowyourhardworkbegins.
Howcanweknowthatthismethodtrisectstheangle?
Anymathematicalproofstartswithsomesetofgivenfacts.Lookagainatthefirstpattypaperimage.Whatdoyouknowforsureaboutthoselines?Makealist!Inordertowritedownwhatyouknow,youwillhavetogivenamestosomeofthepointsandlinesinthedrawing.Eitherprintouttheimageandwriteonit,ormakeasketchofyourown.
Thenlookatthesecondimage.Isthereanythingyouknowforsureaboutthenewlines?Howdoyouknowit?Writedownanyfactsyoucanthinkof,andbesuretoincludeareasonforeachfact.
Nowconcentrateonunderstandingwhatyouaretryingtoprove.Whatexactlydoesitmeantotrisectanangle?Canyouwriteamathematicalstatementorequationthatmeansthesamethingas,Thisangleistrisected?(Youmayneedtolabelsomemorepointsoranglesonyourdiagram.)
Finally,youneedtofindawaytogetfromthefactsyouaregiventothestatementyouwanttoprove,andyouneedtoshowavalidreasonforeachstepalongtheway.Thinkbacktothethingsyoulearnedinmathclass.Whatdoyouknowabouttriangles?Whatdoyouknowaboutparallellines?Willanyofthosefactshelpyouhere?
IfYouGetStumped
IamgoingtoborrowanideafromJonathanatJD2718(http://jd2718.wordpress.com/):thesolutionspost.Iwillwriteafollowuparticlewithafewhints(http://letsplaymath.net/2007/06/05/hintsandsolutionspattypapertrisection/),andthatwillbetheplacetoshareyourquestionsorproofs,ifyouwish.Commentsherebelowarelimitedtogeneralremarksthatwontspoilanyoneelsesfuninworkingthroughthepuzzle.
[Hattip:TrisectinganAnglewithOrigami(http://www.sciencenews.org/view/generic/id/8567/title/Math_Trek__Trisecting_an_Angle_with_Origami)atMathTrek(http://blog.sciencenews.org/view/dispatches/type/column/collection_id/2/title/Math_Trek),viatheCarnivalofMathematics(http://jd2718.wordpress.com/2007/06/02/carnivalofmathematicsix/).]
DontmissanyofLetsPlayMath!: (http://feeds.feedburner.com/LetsPlayMath)Subscribeinareader(http://feeds.feedburner.com/LetsPlayMath),orgetupdatesbyEmail(http://www.feedburner.com/fb/a/emailverifySubmit?feedId=633025&loc=en_US).
2/6/2015 Puzzle:PattyPaperTrisection|Let'sPlayMath!
http://letsplaymath.net/2007/06/05/puzzlepattypapertrisection/ 4/4
HavemorefunonLetsPlayMath!blog:
(http://www.amazon.com/JourneythroughGeniusTheoremsMathematics/dp/014014739X/)
LewisCarrollsLogicChallenges(http://letsplaymath.net/2010/10/11/lewiscarroll%e2%80%99slogicchallenges/)HowtoStartanArgument:TheMontyHallProblem(http://letsplaymath.net/2010/08/16/howtostartanargumentthemontyhallproblem/)LeonhardsBlockPuzzles(http://letsplaymath.net/2008/03/19/leonhardsblockpuzzles/)Puzzle:RandomBlocks(http://letsplaymath.net/2007/07/25/puzzlerandomblocks/)EgyptianGeometryandOtherChallengesandEgyptianMathPuzzles(http://letsplaymath.net/2007/08/24/egyptiangeometryandotherchallenges/)
ByDeniseGaskinsPostedinAlgebra&Beyond,PuzzlesTaggedEuclid,Geometry,Highschool,Mathclub,Proofs,Puzzles
BlogatWordPress.com.|TheiTheme2Theme.
Abouttheseads(http://wordpress.com/abouttheseads/)
Top Related